OVERSAMPLED filter banks have their applications in

Transcription

1 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 46, NO 4, APRIL Oversampled Cosine-Modulated Filter Banks with Arbitrary System Delay Jörg Kliewer, Student Member, IEEE, Alfred Mertins, Member, IEEE Abstract In this paper, design methods perfect reconstruction (PR) oversampled cosine-modulated filter banks with integer oversampling factors arbitrary delay are presented The system delay, which is an important parameter in realtime applications, can be chosen independently of the prototype lengths Oversampling gives us additional freedom in the filter design process, which can be exploited to find FIR PR prototypes oversampled filter banks with much higher stopb attenuations than critically subsampled filter banks It is shown that a given analysis prototype, the PR synthesis prototype is not unique The complete set of solutions is discussed in terms of the nullspace of a matrix operator For example, oversampling allows the design of PR filter banks having unidentical prototypes (of equal unequal lengths) the analysis synthesis stage Examples demonstrate the increased design freedom due to oversampling Finally, it is shown that PR prototypes being designed the oversampled case can also serve as almost- PR prototypes critically subsampled cosine-modulated pseudo QMF banks I INTRODUCTION OVERSAMPLED filter banks have their applications in those areas of signal processing one is interested in making modifications to signals in certain frequency bs Examples are the simulation of room acoustics by filtering in subbs, noise reduction in the spectral domain, equalization via fixed or dynamic (ie time-varying) filtering of subb signals In this paper, we develop PR conditions oversampled cosine-modulated filter banks with arbitrary system delay Especially in real-time applications when there is some feedback involved (ie, echo cancelation mobile phones or hearing aids), it is extremely important that the overall system delay is as low as possible Cosine-modulated filter banks are a special subclass of the general -channel filter bank depicted in Fig 1, the analysis synthesis filters are derived from prototypes, respectively, by cosine modulation Previous work on PR cosine-modulated filter banks [1] [9] has always addressed the critically subsampled case ( in Fig 1) This case is very attractive subb coding because it leads to a minimum number of subb coefficients However, if we simply want to implement some linear filtering by introducing gain factors in the subbs, critical subsampling is not useful This is due to the fact that the main aliasing spectra will no longer cancel in the synthesis filter bank when the subb signals are modified These main aliasing components are not present in the oversampled case, with sufficiently high oversampling Recently, perfect reconstruction (PR) conditions oversampled DFT filter banks have been derived, general relations between oversampled filter banks frame theory have been investigated [10] [13] Due to the modulated nature of DFT cosine-modulated filter banks, the requirements on the prototypes are somewhat related As will be shown the biorthogonal case, all PR prototypes cosine-modulated filter banks also give PR in DFT banks The opposite does not hold all PR DFT prototypes This paper is organized as follows In Section II, we present the polyphase representation general (biorthogonal) oversampled filter banks show how the polyphase matrices in the oversampled case with integer oversampling factors can be obtained from the critically subsampled version Section III derives general PR conditions biorthogonal cosinemodulated filter banks with integer oversampling rates arbitrary overall system delays We show that by solving the PR conditions in the oversampled case, we gain additional design freedom The well-known critically subsampled case the paraunitary oversampled case are regarded as special solutions of our general approach Finally, relations between oversampled cosine-modulated DFT filterbanks will be discussed In Section IV, we establish the connection to pseudo QMF filter banks show that PR solutions in the oversampled case have a partial aliasing cancelation property when applied to a critically subsampled filter bank Section V refers to the prototype design, examples are given Notation: Boldface letters indicate vectors matrices Given a matrix, its transpose is denoted as its Hermitian or transpose-conjugate by, respectively The tilde on a (matrix-) function is defined as, denotes complex conjugation of the polynomial coefficients in The matrices st identity reverse identity matrices, respectively Manuscript received February 15, 1997; revised November 30, 1997 The associate editor coordinating the review of this paper approving it publication was Dr Ahmed Tewfik J Kliewer is with the Institute Network System Theory, University of Kiel, Kiel, Germany A Mertins is with the Department of Electrical Electronic Engineering, University of Western Australia, Nedls, Australia Publisher Item Identifier S X(98) II POLYPHASE REPRESENTATION FOR OVERSAMPLED FILTER BANKS Consider the polyphase representation of the filter bank in Fig 2 Its oversampling ratio is defined as is restricted to be an integer in this paper The analysis polyphase X/98$ IEEE

2 942 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 46, NO 4, APRIL 1998 Fig 1 General analysis synthesis filter bank with subb processing Fig 2 Polyphase analysis synthesis filter bank without subb processing matrix of size in Fig 2 is defined as the polyphase components of the th synthesis filter are components of the (1) are the type-1 polyphase th analysis filter Note that type-2 polyphase components (marked with a bar) are used on the synthesis side However, the description of oversampled modulated filter banks, it will be more convenient to describe the synthesis filter bank with type-1 polyphase components The relation between type-1 type- 2 components can be stated as The synthesis polyphase matrix of size is defined as (2) For critical subsampling, we achieve quadratic polyphase matrices of size, as in the oversampled case, the matrices have a rectangular shape of size, respectively Generally, an -times oversampled filter bank has the perfect reconstruction property if holds, denotes the delay in the polyphase domain Under consideration of the blocking unblocking delay chains in Fig 2, we have an overall delay of samples between We will now derive relations between the polyphase matrices of a given filter bank different (3) (4)

3 KLIEWER AND MERTINS: OVERSAMPLED COSINE-MODULATED FILTER BANKS WITH ARBITRARY SYSTEM DELAY 943 (a) (b) (c) Fig 3 Polyphase analysis filter bank (M =4) (a) Critically subsampled (b) Oversampled by factor 2 scaled by 1= p 2 (c) Equivalent structure to (b) (see text) oversampling ratios, as is restricted to be an integer For this, let us have a look at the example depicted in Fig 3 In Fig 3(a), a critically sampled four-channel analysis filter bank is shown Fig 3(b) shows the same filter bank but with oversampling by the factor a prefactor, which is introduced in order to compensate the increased amplification of the complete analysis synthesis system Note that the polyphase matrix from Fig 3(a) has been replaced by the matrix in Fig 3(b), which means that upsampled versions of the polyphase filters are used Partitioning the polyphase matrix according to rearranging the delay chain the subsamplers yields the structure in Fig 3(c), which is equivalent to the one in Fig 3(b) The system in the gray box is represented by the polyphase matrix of the oversampled system Likewise, the synthesis polyphase matrix can be given as When we assume that the critically subsampled filter bank has the PR property with, the oversampled system also has the PR property Since we neither changed the filters (except by factor ) nor introduced an additional delay into the system, the overall delay of the system must be preserved all oversampling ratios However, the delay caused by the delay chain the oversampled case in Fig 3(c) is reduced by two taps compared with Fig 3(a) Thus, the delay introduced by multiplication of the polyphase analysis synthesis matrix has to be increased by the same amount, which explains the additional delay of in (7) As can be easily verified (same arguments as above), the generalizations of (5) (6) to -channel filter banks being oversampled by are (5) (6) (7) (8) (9) with (10) If the critically sampled filter bank has the PR property (11) then the corresponding oversampled filter bank obviously also guarantees PR, (3) holds This can be easily seen from (8), (9), (11) We have Here, the delay (in the polyphase domain) introduced by the multiplication of the analysis synthesis polyphase matrix amounts to samples a given oversampling ratio The overall delay remains unchanged all, we have III COSINE-MODULATED FILTER BANKS WITH ARBITRARY SUBSAMPLING RATE Unless otherwise noted, we consider biorthogonal cosinemodulated filter banks the analysis filters are derived from an FIR prototype the synthesis filters from an FIR prototype according to The length of the analysis prototype is, the length of the synthesis prototype is denotes the overall delay of the analysis-synthesis system, we will later show that normally varies between the minimal delay of a delay of samples The latter case corresponds to linear-phase prototypes with, which are discussed in Section III-D A suitable choice is given as [2], [3]

4 944 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 46, NO 4, APRIL 1998 In this paper, we restrict ourselves to even analysis synthesis prototype lengths of In this case, all polyphase components of a given prototype have the same length Note that we consider a general approach with different lengths the analysis synthesis prototype A Analysis Synthesis Polyphase Matrices 1) Critically Subsampling: Let denote the type-1 polyphase components of the analysis prototype according to 2) Oversampled Case: We use the description of the polyphase matrices the critically sampled case insert (12) into the relation (8), thus obtaining a more general expression the analysis polyphase matrix in the oversampled case With the definition of diagonal matrices diag this yields the analysis polyphase matrix (17) The analysis polyphase matrix the critically sampled case can be written as [7], [9] (18) (12) The synthesis polyphase matrix can be constructed in the same way From (15) (9), we get (13) (19) is decom- In the same way, the synthesis prototype posed into the polyphase filters (14) Using these type-1 polyphase components, the synthesis polyphase matrix can now be written as (15) the diagonal matrices are defined in the same way as in (14), For completeness, we also give components diag diag (16) in terms of type-2 the matrices containing the type-1 polyphase components of the synthesis prototype are defined as in (17) Note that (compared with critical subsampling) the matrices, respectively, are unchanged in the oversampled case so that the original phase offset in (13) (16) is preserved The matrix is of size,, theree, the polyphase matrices are of size, respectively B PR Conditions In the integer oversampled case, we have two ways of satisfying the PR constraint (3) 1) The PR prototypes the critically subsampled filter bank are also used in the oversampled case, thus, (3) is met all up to the factor 2) When the -times oversampled filter bank is designed in such a way that (3) is satisfied (11) is not, PR up to a constant factor is obtained all oversampling ratios The filter bank designed this way also ensures almost PR all satisfying ; see Section IV The latter case is the interesting one the design of oversampled PR filter banks because we have more degrees of freedom the design process, thus, the conditions on both prototypes can be relaxed Theree, we focus on the discussion of this point First, we derive conditions the polyphase components in order to satisfy (3) arbitrary oversampling ratios

5 KLIEWER AND MERTINS: OVERSAMPLED COSINE-MODULATED FILTER BANKS WITH ARBITRARY SYSTEM DELAY 945 Inserting (18) (19) into (3) yields In order to simplify the expressions (26) (27), we first write (26) as (20) we use the type-1 polyphase description on the synthesis side Choosing as in (13), as in (16), writing the delay as leads to the expression the product [9], [14] (21) (28) Note that (28) has the m of a polyphase decomposition of some filter (which is identically zero here), theree, we can conclude that (28) can be written as a set of independent equations Inserting this product into (20), we get (22) (23) (29) Here, was replaced by Equation (27) can be simplified by replacing the argument with by writing the delay with (4) (21) as (30) For achieving the PR property, the second term of this equation, which contains the antidiagonal terms, must be zero With (18) (19), this can be written as thus resulting in (31) (24) When this condition is satisfied, we still must constrain the first term of (23) to be a simple delay With (18), (19), (23), we then can state the requirement (25) We now use (17) express the two PR conditions (24) (25) directly with the polyphase components This gives the conditions To achieve a PR filter bank, we have to fulfill (29) find a feasible solution that also satisfies (31) We will see in the next sections that we have different solutions in the critically sampled the oversampled case, in the latter case, we have less severe constraints the analysis synthesis prototypes, which will give us some benefits the prototype filter design The overall length of filters of the type in (31) amounts to taps, which allows the delay parameter to vary between 0 Thus, can be understood to be a design parameter that allows us to choose the overall delay of the analysis-synthesis system independently of We can achieve minimum delay prototypes linear-phase prototypes (with ) which have to be fulfilled (26) (27) C General Solutions the Design of PR Prototypes We can see that (29) is always fulfilled if the synthesis prototype the analysis prototype are related as with ; they differ only by a constant factor an additional delay However, in general, we have more solutions more design freedom In order to describe the complete set of solutions the oversampled case, we rewrite the linear set of (29) (31) as (32)

6 946 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 46, NO 4, APRIL 1998 The matrix of dimension has a block diagonal structure identical analysis synthesis prototypes, (29) is always satisfied Then, all solutions can be characterized by Because of this block structure, we can split the whole system into smaller linear sets of equations The following partitioning of the vectors is used (33) (35) the coefficients denote the free parameters in the above problem The terms are arbitrary (FIR or IIR) transfer functions Additionally, we may introduce further delay such that Note that the role of the analysis synthesis prototype can be interchanged This can be seen from the PR constraints in (29) (31), which still hold when plays the role of vice versa Thus, instead of (35), we can also write The submatrices have the dimension With the abbreviation, they can be written as (34), shown at the bottom of the page Note that the first row of (33) is due to (31), as the remaining rows are due to (29) Since the matrices are rectangular of size, we have underdetermined sets of equations with at least free parameters All solutions to (32) can be obtained by taking one solution of the inhomogeneous system adding linear combinations of all other solutions from the nullspace of (these are the solutions of ) For the sake of simplicity, let us assume that the matrixes have maximal rank, that is, rank (otherwise, we would have polyphase components with zero coefficients) This means we obtain linearly independent basis vectors each nullspace Such a set of basis vectors the nullspace will be denoted as with These considerations show that we can construct PR filter banks with different analysis synthesis prototypes The synthesis filters can be longer or shorter as the analysis filters, in special cases, we can also have unidentical prototypes of the same length This is demonstrated in the example below Obviously, if we decompose a signal with our analysis filter bank we only use a nullspace filter as a synthesis prototype, the output signal will be identically zero If we construct the synthesis prototype according to (35), the output will be independent of the parameters of the filters However, it is important to notice that the output signal only is independent of the nullspace component as long as the subb signals are directly derived from the analysis filter bank If the subb signals are modified, which is the case in real applications, the nullspace component will also influence the output Theree, the nullspace filters can be used in order to modify optimize the prototypes Example: Let us consider the case The matrix is then given by the equation at the bottom of the next page, the basis vectors the nullspace of can be obtained as sts the polyphase components of the corresponding nullspace filter When we design the analysis prototype in such a way that it satisfies (31) with, its polyphase components serve as a special solution to (33) because (34)

7 KLIEWER AND MERTINS: OVERSAMPLED COSINE-MODULATED FILTER BANKS WITH ARBITRARY SYSTEM DELAY 947 From this expression, we see that the nullspace polyphase components have a maximum length of when the analysis polyphase components are of length With (35), this leads to analysis synthesis prototypes of unequal length However, if we have the same lengths both prototypes, but the analysis synthesis prototypes do not have to be identical These conditions can be satisfied by designing the analysis prototype the critically subsampled case [according to (31) with ] When we scale this analysis prototype by apply it to an oversampled filter bank with, we can construct the corresponding PR synthesis prototypes via (35) In this case, the two nullspace basis vectors reduce to with, we obtain unidentical analysis synthesis prototypes of the same length Fig 4 shows an example the nullspace solution the analysis prototype of length is designed The magnitude frequency response of the analysis prototype is depicted in Fig 4(a) For the choice the frequency response of the nullspace filter is shown in Fig 4(b) Thus, the nullspace impulse response has a length of Note that the passb of the nullspace filter is located within the stopb of the analysis prototype This gives a vivid explanation of the properties of a nullspace filter However, since we have the free choice, such a behavior must not be found in all nullspace filters A synthesis prototype was obtained as the sum of the analysis the nullspace filter Fig 4(c) depicts the corresponding magnitude frequency response, which qualitatively looks like the sum of the responses in Fig 4(a) (b) A more constructive solution is depicted in Fig 4(d) In this case, the polyphase components of the analysis the nullspace filters are linearly combined in such a way that the stopb attenuation of the resulting synthesis filter is increased D PR Solution Special Cases 1) Critically Subsampled Case: In the following, it is shown how the results the critically subsampled case as stated in literature [7], [9] are related to the general solution presented in the last section With, (31) (29) can be written as (36) (37) In (36), two products of analysis synthesis prototype polyphase components always have to add up to a delay, which is more restrictive than (31) in the oversampled case The submatrices in (34) now have the dimension, the individual systems are given as Since is now quadratic is assumed to have full rank, the nullspace is just the null vector This means that given the analysis prototype the delay parameter,we have only one solution the synthesis prototype, which can be written in closed m as An FIR solution the synthesis filter is obtained if which is exactly the same condition as (36) with In other words, an FIR solution the case essentially requires analysis synthesis prototypes with poyphase components being equal up to a scaling factor 2) Paraunitary Case: The paraunitary case is characterized by the fact that the sum of the energies of all subb signals is equal to the energy of the input signal This may be expressed as with, is the polyphase vector of a finite-energy input signal, is the vector of subb signals denotes the Euclidean norm As can be easily verified, filter banks (oversampled critically sampled) are paraunitary if (38)

8 948 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 46, NO 4, APRIL 1998 (a) (b) (c) Fig 4 Examples magnitude frequency responses (M =4;N =2;D1 =0) (a) Analysis prototype (b) Nullspace filter ( (0) 0 = (0) 1 =20; C0(z)= C1(z)=1) (c) Synthesis prototype according to (35) with (0) 0 = (0) 1 =20;C0(z)= C1(z)=1 (d) Synthesis prototype according to (35) with (0) 0 = 01; (0) 1 =1;C0(z)= C1(z)=1: (d) holds From this equation, we may conclude that (39) For identical linear-phase analysis synthesis prototypes, from (40) (31), it follows that yields a PR synthesis filter bank The delay in (39) is introduced in order to achieve causal synthesis filters The polyphase matrix contains the solutions from the nullspace, which have been introduced in Section III-C In the following, we restrict ourselves to From (39), we then see that the synthesis filters are timereversed versions of the analysis filters, that is, As in the critically sampled case [14], this is fulfilled when we choose linear-phase prototypes For the overall delay, this means that is fixed to The delay parameters can be identified as, respectively Due to the linear-phase property of its length being an integer multiple of, two polyphase components are always related as (40) (41) This result was already established in [15] Equation (41) states that polyphase components always have to be power complementary in the oversampled case For increasing, this condition on the prototype becomes less restrictive, the design freedom increases Note that (41) is the only condition on the analysis prototype paraunitary oversampled filter banks because with, the condition (29) is always fulfilled E Relation to DFT Filter Banks In DFT filter banks, the analysis synthesis filters,, respectively, are obtained by complex modulation from given prototypes The literature [10], [11], [13] covers only the paraunitary case, the same prototype is used analysis synthesis However, relating DFT filter banks to the cosine-modulated banks derived in this paper, it is useful to consider two different prototypes ( )

9 KLIEWER AND MERTINS: OVERSAMPLED COSINE-MODULATED FILTER BANKS WITH ARBITRARY SYSTEM DELAY 949 also DFT banks Thus, Note that we consider -b DFT -b cosine-modulated filter banks so that the subbs are of equal spectral width in both cases In the critically subsampled case, the analysis synthesis polyphase matrices the DFT filter bank can be written with as in (21) the diagonal matrix with elements as oversampling ratio as an almost PR solution with These solutions have the interesting property that every th aliasing spectrum is canceled 1) Partial Aliasing Cancelation: We first state some general results consider an arbitrary -channel filter bank with oversampling ratio subb sampling rate,as it is depicted in Fig 1 The analysis modulation (which is also called the aliasing component) matrix of size is defined as diag diag (42) (43) Here, denotes the DFT matrix the the type-1 polyphase components of the analysis synthesis prototype, respectively Similar to the derivation the cosine-modulated filter bank in Section III-B, the PR constraints the -times oversampled DFT filter bank with can be obtained by inserting (42) into (8), (43) into (9), replacing by Hence, the PR condition in (3) can be written as We now define the transfer functions (45) (44) which is a similar expression as (20) in the cosine-modulated case half the oversampling ratio However, since, it turns out that it is sufficient to satisfy only the following PR condition similar to (31) in order to achieve PR an -times oversampled DFT filter bank Due to the absence of antidiagonal terms in the product of the analysis synthesis transm matrices [compare (22)], the additional PR conditions the cosine-modulated case in (29) are not relevant in the oversampled DFT case Thus, all PR solutions the -times oversampled cosine-modulated case including unidentical analysis synthesis prototypes can also be used as solutions in the -times oversampled DFT case, but they only represent a subset of all possible solutions All considerations regarding the system delay, the link to pseudo QMF filter banks, the design issues in the next sections can be applied to oversampled DFT filter banks as well which denote the aliasing contribution of the analysis filter shifted by ( ) the overall transfer function of the nonsubsampled filter bank The aliasing component vector can be obtained by (46) with the synthesis filter vector On the other h, we can express the aliasing component matrix (45) in terms of the analysis polyphase matrix (1) according to [16] as (47) diag, denotes the DFT matrix The synthesis filters can be constructed from their type-2 polyphase components according to Writing this with the synthesis polyphase matrix (2) yields (48) Combining (46) (48) leads to IV RELATION TO PSEUDO QMF FILTER BANKS The approaches presented in the previous sections can not only be used to design PR -times oversampled filter banks As we will show in the following, it is also possible to use a prototype that satisfies the PR constraints a given (49) Now, we use the fact that is equal to the left-h side of (25) if (29) is fulfilled (eg, because of the choice ) By inserting this relation into (49)

10 950 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 46, NO 4, APRIL 1998 by expressing the resulting equation with the polyphase components,weget (50) The comparison of (50) with (31) shows that the PR case, each sum over in (50) amounts to a delay Then, it can be shown easily that, denotes the overall system delay This means that in the PR case, all aliasing components are canceled at the summation point in the synthesis filter bank In order to describe the aliasing-cancelation effects in the pseudo QMF case, we use PR prototypes designed in an -times oversampled filter bank, we write (50) with an additional scaling factor as In the last step, the sum over was split into two sums by the substitution, is the subb decimation factor which the prototype filters are designed We now substitute again, which yields (51) For those is a multiple of, the term in (51) becomes independent of, then, the inner sum corresponds to the PR constraint (31) an oversampling ratio When we use prototypes, which give PR [ie, satisfy (31)], the inner sum can be replaced by the delay term, we have Thus, every th aliasing component is canceled out The overall transfer function amounts to, the overall system delay (again) can be found as Furthermore, we have to scale the analysis synthesis prototype with factor in order to obtain the original amplitude With such scaling, we have no linear distortion at all, which is not the case when using prototype filters designed as approximate Nyquist filters almost PR filter banks [17] [19] The only output distortion is due to those aliasing components that are not canceled 2) Examples Pseudo QMF Solutions: Generally, the filter bank in Fig 1 can be regarded as a periodically timevarying system with the periodic system response, denotes the response of the system at discrete time to a unit sample applied at discrete time The aliasing distortions in such systems are best visualized via bifrequency system functions [20] [22] The bifrequency system function is defined as [22] Since is periodic in contains only discrete lines, which refer to the aliasing components the overall frequency response Fig 5(a) shows the magnitude bifrequency system function the PR oversampled case with subbs, an oversampling ratio of, a prototype of length designed The diagonal lines indicate the principal location of all possible aliasing components, which are completely canceled due to the PR property, only the line the transfer function of the system remains Since we have a one-toone mapping between the input frequency the output frequency, this system can be regarded as time invariant The PR prototype is now scaled by applied to a critically subsampled filter bank, the resulting system bifrequency function is shown in Fig 5(b) We can see that every second possible aliasing spectrum is zero, which confirms the result of the proof above Since the remaining aliasing components are significantly suppressed by the high stopb attenuation of the subb filters by the property of the cosine-transm in (13), respectively, this case can be regarded as the pseudo QMF case However, since the inputoutput mapping is not one-to-one, the system response of this filter bank is not time invariant In a second example, we apply a prototype of length designed to a critically subsampled filter bank with This case can be regarded as the classical pseudo QMF case, all aliasing components are present with a magnitude that corresponds to the prototype filter s stopb attenuation The resulting bifrequency system function is shown in Fig 6 Note that although we have chosen, the aliasing spectrum is not present This is due to the even length of the prototype filter, which leads to a zero at in the transfer function, thus, to a suppression of all signal components (including aliasing) at the frequency

11 KLIEWER AND MERTINS: OVERSAMPLED COSINE-MODULATED FILTER BANKS WITH ARBITRARY SYSTEM DELAY 951 (a) (b) Fig 5 Normalized magnitude system bifrequency functions (a) Oversampled perfect reconstruction case M = 8; L =2 Lo =2 (b) Critically sampled case with M =8 the same prototype as in (a) The prototype filter has a length of Lp = 128: Expressing the sums results in the requirements as impulse responses (53) Fig 6 Normalized magnitude system bifrequency function a critically sampled filter bank with M =8 a prototype of length 128 designed Lo = M: V DESIGN PROCEDURE AND EXAMPLES A Prototype Optimization 1) General Case: We have seen in Section III-C that in the case of identical prototypes the analysis synthesis bank, (29) is satisfied, the prototype, according to (31), remains to be designed Expressing the latter equation in the time domain with yields this yields the PR constraints in the time domain otherwise (54) The delay parameter can be selected in the range The number of PR constraints in (54) is For being a power of two, this means a reduction by factor compared with the critically subsampled case For the design of the prototype, we use a quadraticconstrained least-squares approach [23] Each from (54) can be written as a quadratic constraint of the m with some matrix For details on the optimization approach, see [23] The constraints (54) are fulfilled numerically under additional minimization of the weighted prototype s stopb energy, which is independent of can be written as (55) (52), denotes the unit sample sequence The sequences st the result of the convolution of the polyphase impulse responses Since we consider an analysis prototype filter of length, each polyphase component has the length, which results in a total convolution length of The index can be restricted to the first convolution results because the first results lead to exactly the same expressions as the last ones The case of odd is mally taken into account by use of the ceiling operator Here, denote the weighting factors the different frequency regions, which are characterized by the edge frequencies Equation (55) can be expressed as a quadratic m according to with

12 952 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 46, NO 4, APRIL 1998 (a) (b) Fig 7 Design Example 1 Magnitude frequency responses linear-phase prototypes Design parameters: L p = 256;M =16;D 1 =7;Q =2;edge frequencies [ s ; s ; s ]=[0:06; 0:3; ], weights w 0 =1;w 1 =2; = : (a) Oversampled case with L =2 (b) Critically sampled case Note that since is a real vector, we only need to take the real part of the outer product The elements of the matrix can be given analytically as Re 2) Paraunitary Case: In this special case with linear-phase prototypes satisfying the symmetry condition (40), we have only time-domain constraints to be satisfied in order to achieve PR It was already stated in Section III-D that linear-phase filters, the delay parameter is fixed to so that we can write (52) as The expressions can now be regarded as the autocorrelation sequences of the polyphase impulse responses shifted by samples Due to the even symmetry of the autocorrelation, we can restrict ourselves to Hence, in the paraunitary case, we have time-domain PR constraints according to with defined as in (53) This is a further reduction of the required conditions compared with the general case above When comparing the number of PR constraints different oversampling ratios, here, we obtain the same results as in the general case B Design Examples In the following, we present design examples, which show that by relaxing the number of constraints in the oversampled case, we can design filters with better properties In all examples, the initial filters the optimization process were constructed via the pseudo QMF method from [19] As an objective function, we used the stopb energy (55) The optimization was carried out with the NAG Fortran library The PR constraints (54) were fulfilled up to some accuracy, which will be specified in the examples For displaying frequency responses, the prototypes were normalized to 0 db DC amplification Example 1: In this example, we address the paraunitary case design a linear-phase prototype filter of length subbs an overall system delay of samples Fig 7(a) shows the magnitude frequency response of the prototype designed the oversampled case with, Fig 7(b) corresponds to the critically subsampled case The oversampled case yields much higher stopb attenuation, which shows that the additional design freedom in the oversampled case can indeed be used to improve the prototype Example 2: Here, we refer to the biorthogonal case consider an example with reduced delay, a prototype of length is designed, an overall system delay of samples an oversampling ratio of The prototype is then compared with a linear-phase design with the same reconstruction delay The solid line in Fig 8(a) shows the magnitude frequency response of the low-delay prototype with, the dotted line corresponds to the linear-phase case with As we can see, the longer prototype yields higher stopb attenuation while having the same reconstruction delay Fig 8(b) shows the magnitude frequency response the critically subsampled low-delay case with the same parameters as above We see again that in the biorthogonal case as well, the prototypes do not have linear phase, we gain advantages by designing prototypes especially a given oversampling ratio

13 KLIEWER AND MERTINS: OVERSAMPLED COSINE-MODULATED FILTER BANKS WITH ARBITRARY SYSTEM DELAY 953 (a) (b) Fig 8 Design Example 2: Magnitude frequency responses M = 8 an overall system delay of D = 47; one frequency b in the stopb region with s = 0:1; = (a) Oversampled case L = 2 with L p = 128 (solid line) L p = 48 (linear-phase case, dotted line) (b) Critically sampled case with L p = 128: (a) (b) Fig 9 Design Example 3: Prototype length L p = 512 M = 32subbs, oversampling ratio L = M, an overall system delay of D = 447 samples (D 1 = 6;Q = 1; s = 0:03; = ) (a) Magnitude frequency response (b) One period of the filter bank s overall magnitude frequency response ja 0 (e j )j: Example 3: In this example, we design a prototype the nonsubsampled case of length an overall system delay of samples Fig 9 shows the filter bank s magnitude frequency response one period of the overall magnitude frequency response Due to the high stopb attenuation, this prototype can be used a critically subsampled pseudo QMF filter bank, the aliasing components in the stopb region of the prototype are suppressed The main aliasing components, however, are canceled out by the properties of the transms (13) (16) 1) Relations Between Stopb Energy, Stopb Attenuation, Filter Length, Oversampling Ratio: In the following, we show the effects on the stopb energy the stopb attenuation in the paraunitary case when the filter length the oversampling ratio are varied The results are displayed in Fig 10 All prototypes are designed with, respectively Due to the quadratic nature of the objective function (55), the fact whether or not the obtained solutions correspond to local or global minima highly depends on the size of the problem on the optimization method used As expected, we can observe that the stopb energy decreases when the filter length increases; see Fig 10 However, in the case filter lengths larger than 350, there is not much improvement, one encounters the limits of the used optimization algorithm There are even cases the best filters have been found critical subsampling Since these prototypes also give PR in the oversampled case, we can conclude that in these cases, the solutions represent local minima of the objective function From Fig 10(a) (b), we see that fixed prototype lengths, the stopb energy decreases when the subsampling rate becomes smaller This confirms the observation in Example 1 A decreasing leads to a decreasing number of PR constraints, thus, we may expect a lower stopb energy However, oversampling ratios bring only negligibly small improvements compared with For the stopb attenuation, which is the interesting measure in practice, we qualitatively have the same results as the stopb energy See Fig 10(c) (d) average stopb attenuations 2) Relations Between Stopb Energy, Stopb Attenuation, System Delay, Oversampling Ratio: We here address

14 954 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 46, NO 4, APRIL 1998 (a) (b) (c) Fig 10 Stopb energy (a) M = 16 (b) M =32 average stopb attenuation (c) M =16 (d) M =32 versus prototype length different subsampling rates (d) (a) (b) Fig 11 Stopb energy (a) average stopb attenuation (b) versus overall system delay different subsampling rates (M = 8;Lp = 128): the biorthogonal case measure the effects on the stopb energy the stopb attenuation due to a change of the parameter, thus, due to the overall system delay different oversampling ratios The other design parameters are fixed; they are The results are shown in Fig 11 We observe that increasing delay, the objective function decreases As in the previous example, we see that an oversampling ratio of

15 KLIEWER AND MERTINS: OVERSAMPLED COSINE-MODULATED FILTER BANKS WITH ARBITRARY SYSTEM DELAY 955 with only gives a small amount of improvement compared VI CONCLUSION We have derived the PR conditions oversampled cosinemodulated filter banks with arbitrary system delay It turned out that given an analysis prototype, the synthesis prototype is not uniquely determined The complete set of solutions was expressed in terms of a single solution the solutions from the nullspace of a matrix operator Within this framework, PR filter banks with unidentical prototypes the analysis synthesis stage can be designed The design examples have shown that PR prototypes oversampled filter banks with much higher stopb attenuations than critically subsampled filter banks can be found All PR solutions in the cosine-modulated case can be used oversampled DFT filter banks as well Finally, relations between PR oversampled critically subsampled cosine-modulated pseudo QMF banks were discussed It was shown that pseudo QMF banks having a partial aliasing-cancelation property can be designed via the design of PR oversampled filter banks REFERENCES [1] T A Ramstad J P Tanem, Cosine-modulated analysis synthesis filterbank with critical sampling perfect reconstruction, in Proc IEEE Int Conf Acoust, Speech, Signal Process, Toronto, Ont, Canada, May 1991, pp [2] R D Koilpillai P P Vaidyanathan, Cosine-modulated FIR filter banks satisfying perfect reconstruction, IEEE Trans Signal Processing, vol 40, pp , Apr 1992 [3] H S Malvar, Signal Processing with Lapped Transms Norwood, MA: Artech House, 1992 [4] J Mau, Perfect reconstruction modulated filter banks: Fast algorithms attractive new properties, in Proc IEEE Int Conf Acoust, Speech, Signal Process, Minneapolis, MN, Apr 1993, vol 3, pp [5] R A Gopinath C S Burrus, Theory of modulated filter banks modulated wavelet tight frames, in Proc IEEE Int Conf Acoust, Speech, Signal Process, Minneapolis, MN, Apr 1993, vol 3, pp [6] R Gluth, A unified approach to transm-based FIR filter banks with special regard to perfect reconstruction systems, in Proc IEEE Int Conf Acoust, Speech, Signal Process, Minneapolis, MN, Apr 1993, vol 3, pp [7] T Q Nguyen P N Heller, Biorthogonal cosine-modulated filter banks, in Proc IEEE Int Conf Acoust, Speech, Signal Process, Atlanta, GA, May 1996, pp [8] G D T Schuller M J T Smith, New framework modulated perfect reconstruction filter banks, IEEE Trans Signal Processing, vol 44, pp , Aug 1996 [9] P N Heller, T Karp, T Q Nguyen, A mulation of modulated filter banks, submitted publication [10] Z Cvetković, Oversampled modulated banks tight Gabor frames in `2(Z), in Proc IEEE Int Conf Acoust, Speech, Signal Process, Detroit, MI, May 1995, pp [11] Z Cvetković M Vetterli, Oversampled filter banks, submitted publication [12] M Vetterli Z Cvetković, Oversampled FIR filter banks frames in `2(Z), in Proc IEEE Int Conf Acoust, Speech, Signal Process, Atlanta, GA, May 1996, pp [13] H Bölcskei, F Hlawatsch, H G Feichtinger, Oversampled FIR IIR DFT filter banks Weyl-Heisenberg frames, in Proc IEEE Int Conf Acoust, Speech, Signal Process, Atlanta, GA, May 1996, pp [14] P P Vaidyanathan, Multirate Systems Filter Banks Englewood Cliffs, NJ: Prentice-Hall, 1993 [15] J Kliewer A Mertins, Design of paraunitary oversampled cosinemodulated filter banks, in Proc IEEE Int Conf Acoust, Speech, Signal Process, Munich Germany, Apr 1997, pp [16] M Vetterli, Multirate filter banks subb coding, in Subb Image Coding, J W Woods, Ed Boston, MA: Kluwer, 1991, pp [17] P L Chu, Quadrature mirror filter design an arbitrary number of equal bwidth channels, IEEE Trans Acoust, Speech, Signal Processing, vol ASSP-33, pp , Feb 1985 [18] C D Creusere S K Mitra, A simple method designing highquality prototype filters M-b pseudo QMF banks, IEEE Trans Signal Processing, vol 43, pp , Apr 1995 [19] J Kliewer, Simplified design of linear-phase prototype filters modulated filter banks, in Proc Euro Signal Process Conf, Trieste, Italy, 1996, pp [20] R L Reng H W Schuessler, Measurement of aliasing distortions quantization noise in multirate systems, in Proc IEEE Int Symp Circuits Syst, San Diego, CA, 1992, pp [21] F A Heinle H W Schuessler, An enhanced method measuring the permance of multirate systems, in Proc Int Conf Digital Signal Process, Limassol, Cyprus, June 1995 [22] R E Crochiere L R Rabiner, Multirate Digital Signal Processing Englewood Cliffs, NJ: Prentice-Hall, 1983 [23] T Q Nguyen, Digital filter banks design Quadratic-constrained mulation, IEEE Trans Signal Processing, vol 43, pp , Sept 1995 Jörg Kliewer (S 97) was born in Hamburg, Germany, on January 23, 1966 He received the Diploma degree in electrical engineering from the Hamburg University of Technology, Hamburg, Germany, in 1993 In December 1993, he joined the University of Kiel, Kiel, Germany, as a Research Assistant, he is currently pursuing the PhD degree In 1996, he spent four months at the University of Calinia, Santa Barbara, as a visiting researcher His research interests include multirate filtering, audio image source coding, filter design Alfred Mertins (M 96) received the Diplomingenieur degree from the University of Paderborn, Paderborn, Germany, in 1984 the Dr-Ing degree in electrical enginering the Dr-Ing habil degree in telecommunications from the Hamburg University of Technology, Hamburg, Germany, in , respectively From 1986 to 1991, he was a Research Associate in the Telecommunications Institute, Hamburg University of Technology From 1991 to 1995, he was with the Microelectronics Applications Center, Hamburg, Germany, as a Senior Research Associate From 1996 to 1997, he was a Senior Lecturer at the University of Kiel, Kiel, Germany, he gave courses on telecommunications, stochastic signal analysis, multirate digital signal processing wavelets In October 1997, he joined the Visual Communication Research Group, Department of Electrical Electronic Engineering, University of Western Australia, Nedls, Australia, as a Senior Research Fellow His research interests include image video processing, wavelets filter banks, joint source-channel coding, digital communications